Can anybody explain me the relation between the condition of a Matrix and the convergency of a problem.
For example how is the relation between the condition of the stiffness Matrix occuring in FEM to the convergency of the FEM method?

Thank you!

Note: For example I want to solve problem In weak formulation: find $u\in H^1_{\Gamma_D}(\Omega)$ for $f\in L^2(\Omega)$, $g_D\in H^{\frac{1}{2}}(\Gamma_D)$ and $g_N \in L^2(\Gamma_N)$ such that for all $v\in H^{1}_0$

$\begingroup$What kind of matrix? Jacobian, gradient, Hessian or something else? Optimize or solve system of equation? Are there any constraints?$\endgroup$
– R zuOct 28 '18 at 17:33

$\begingroup$The Stiffness matrix which occurs for example using FEM for an elliptic linear PDE. Maybe i should make a clearer post with more details. Let me change it in the o.g. post.$\endgroup$
– KeremOct 28 '18 at 17:39

$\begingroup$Hi welcome to scicomp, if I understand your question, I suppose it is about the convergence of linear solver for the system $Au=f$ (different from the convergence of FEM). If is so maybe you are using things like coniugate gradient (CG) o precond coniugate gradient (PCG). If I am correct place tell us (and modify the question.$\endgroup$
– Mauro VanzettoOct 28 '18 at 17:59

1 Answer
1

I suppose there is an misunderstanding about the convergence, in my opinion it is about the convergence of the linear solver used to solve the linear system $Au=f$ and no the convergence of the method FEM itself. I try to explain.

h is the discretization index i.e. the grid step. More $h$ is small more $V_h$ is big so you have got a best approximation.

The motivation about the Galerking method works is essentially based on the Céa's lemma, it give us this estimate:
$$
||u -u_h||_V \leq c \inf_{v \in V_h}||u - v||_V
$$

In the wiki page there is an example of estimate with the approx by piecewise-linear function.

The choice of the base for the space is important, in the early time before computers often were used global or trigonometric polynomial.

When you choose piecewise polynomial, i.e simple polinomial with compact support, you obtain FEM method. For example it has got the advantage that the stiff matrix is sparse.

As you can see form the Céa's lemma and this estimates:
$$
||u - u_h||_{0,\infty} \leq C h^{l+1} |u|_{l+1, \infty} \; \forall u \in W^{l+1, \infty}
$$
, here $W^{l+1, \infty}$ is a Sobolev's space, the convergence of the FEM method is related to the step $h$. For this I think that your question is about the second part.

Linear solver convergence

If you have got:

data with infinite precision

solve the linear system with infinite precision. For example by hand gauss elimination with infinite precision, or conjugate gradient (CG) that with infinite precision is an exact solver and converge after $n = dim(A)$ iteration
you are immune about $cond(A)$.

But as the Spartans said, if...
For point one you can not meet this requirement.
For point 2, it is impractical, to much time by hand and also if you want use CG and wait the final step the dimension of $A$ give you problem.

For this is important to consider $cond(A)$ because influence your convergence.

For example, only to show a case, if you consider CG that is often used yo have got this estimate: